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Abstract

Our computations show that there is a total of $40$ pairs of degree six
coprime polynomials $f,g$ where $f(x)=(x-1)^6$, $g$ is a product of cyclotomic
polynomials, $g(0)=1$ and $f,g$ form a primitive pair. The aim of this article
is to determine whether the corresponding $40$ symplectic hypergeometric groups
with a maximally unipotent monodromy follow the same dichotomy between
arithmeticity and thinness that holds for the $14$ symplectic hypergeometric
groups corresponding to the pairs of degree four polynomials $f,g$ where
$f(x)=(x-1)^4$ and $g$ is as described above. As a result we prove that at
least $18$ of these $40$ groups are arithmetic in $\mathrm{Sp}(6)$.
In addition, we extend our search to all degree six symplectic hypergeometric
groups. We find that there is a total of $458$ pairs of polynomials (up to
scalar shifts) corresponding to such groups. For $211$ of them, the absolute
values of the leading coefficients of the difference polynomials $f-g$ are at
most $2$ and the arithmeticity of the corresponding groups follows from Singh
and Venkataramana, while the arithmeticity of one more hypergeometric group
follows from Detinko, Flannery and Hulpke.
In this article, we show the arithmeticity of $160$ of the remaining $246$
hypergeometric groups.